Irreducibility of Multivariable Polynomials

Question

Question: Let $k$ be a field and $p,q\in k[x]$ two relatively prime polynomials. Prove $p(x)y-q(x)$ is irreducible in $k[x,y]$.

How does one show this? More generally, how does one show that multivariable polynomials are irreducible? In one variable we have access to tools like Gauss's lemma and Eisenstein's criteria, but I do not know any methods applicable to the multivariable case.

Answer 1

Define $f(x,y) = p(x) y - q(x)$. Suppose $f(x,y) = g(x,y) h(x,y)$.

In the ring $k(x)[y]$, $f(x,y)$ is clearly irreducible; therefore one of $g(x,y)$ and $h(x,y)$ is a unit. WLOG, assume $g(x,y)$ is the unit. The only units in $k(x)[y]$ are the nonzero elements of $k(x)$, therefore $g(x,y) = e(x)$ for some $e$.

If $f(x,y) = e(x) h(x,y)$, then $e(x)$ must be a common divisor of $p(x)$ and $q(x)$; therefore $e(x)$ is a unit of $k[x]$, and thus $g(x,y)$ is a unit of $k[x,y]$.

Answer 2

All multivariate polynomials are also one-variable polynomials. This polynomial is a polynomial in $y$ with coefficients in $k[x]$, which is a UFD, so Gauss's lemma and Eisenstein's criterion continue to apply (but are unnecessary because here the polynomial is linear...).